Prove the following trigonometric identities:$ (\operatorname{cosec} A-\sin A)(\sec A-\cos A)(\tan A+\cot A)=1 $

To do:

We have to prove that \(  (\operatorname{cosec} A-\sin A)(\sec A-\cos A)(\tan A+\cot A)=1 \).

Solution:

We know that,

$\tan A=\frac{\sin A}{\cos A}$.....(i)

$\cot A=\frac{\cos A}{\sin A}$.....(ii)

$\operatorname{cosec} A=\frac{1}{\sin A}$.....(iii)

$\sec A=\frac{1}{\cos A}$.....(iv)

$\sin^2 A+\cos ^{2} A=1$.......(v)

Therefore,

$(\operatorname{cosec} A-\sin A)(\sec A-\cos A)(\tan A+\cot A)=(\frac{1}{\sin A}-\sin A)(\frac{1}{\cos A}-\cos A)(\frac{\sin A}{\cos A}+\frac{\cos A}{\sin A})$ 

$=(\frac{1-\sin^2 A}{\sin A})(\frac{1-cos^2A}{\cos A})(\frac{\sin^2 A+\cos^2A}{\sin A\cos A})$ 

$=(\frac{\cos^2A}{\sin A})(\frac{sin^2A}{\cos A})(\frac{1}{\sin A\cos A})$ 

$=\frac{\sin A\cos A}{\sin A\cos A}$                

$=1$

Hence proved.    

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Updated on: 10-Oct-2022

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